warm-up: january 12, 2012 find all zeros of. homework questions?
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Warm-Up: January 12, 2012 Find all zeros of
1243 23 xxxxf
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Homework Questions?
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Rational Functionsand Their Graphs
Section 2.6
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Objectives1. Find the domain of rational functions2. Use arrow notation3. Identify vertical asymptotes4. Identify horizontal asymptotes5. Graph rational functions6. Identify slant asymptotes7. Solve applied problems involving rational
functions
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Rational Functions Rational Functions are quotients of
polynomial functions
The domain of a rational function is all real numbers except those that cause the denominator to equal 0
xqxp
xf
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Example 1 (like HW #1-8) Find the domain of
64
82
x
xxf
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You-Try #1 (like HW #1-8) Find the domain of
2
52
xx
xxf
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Arrow Notation As x a+, f(x) ∞
“As x approaches a from the right, f(x) approaches infinity”
As x a-, f(x) -∞ “As x approaches a from
the left, f(x) approaches negative infinity”
As x ∞, f(x) 0 “As x approaches
infinity, f(x) approaches zero”
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Vertical Asymptotes An asymptote is a line that the graph of f(x)
approaches, but does not touch. The line x=a is a vertical asymptote if f(x)
increases or decreases without bound as x approaches a. As x a+, f(x) ±∞ As x a-, f(x) ±∞
If “a” is a zero of q(x), but not a zero of p(x), then x=a is a vertical asymptote.
xqxp
xf
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Example 2 (like HW #21-28) Find the vertical asymptotes, if any, of
xx
xxf
52
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You-Try #2 (like HW #21-28) Find the vertical asymptotes, if any, of
16
42
x
xxf
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Holes A hole is a point that is not part of the domain
of a function, but does not cause an asymptote.
If “a” is a zero of q(x), and a zero of p(x), then there is a hole at x=a
Holes generally are not distinguishable on a graphing calculator graph
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Example of a Hole
2,22
222
42
xxxfx
xxxf
x
xxf
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Horizontal Asymptotes The line y=b is a horizontal asymptote if
f(x) approaches “b” as x increases or decreases without bound As x ∞, f(x) b OR As x - ∞, f(x) b
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Identifying Horizontal Asymptotes Only the highest degree term of the top and
bottom matter
Let “n” equal the degree of p(x), the numerator
Let “m” equal the degree of q(x), the denominator
If n<m, then the x-axis (y=0) is the horizontal asymptote
If n=m, then the line is the horizontal asymptote
If n>m, then f(x) does not have a horizontal asymptote
...
...
m
m
nn
xb
xa
xq
xpxf
m
n
b
ay
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Example 3 (like HW #29-33) Find the horizontal asymptote, if any, of each
function
23
6
23
6
23
62
3
2
2
2
x
xxh
x
xxg
x
xxf
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Warm-Up: January 17, 2012 Find the horizontal asymptotes, if any, of:
Find the vertical asymptotes, if any, of
34
23
23
3
2
3
7
23
7
23
76
23
xx
xxxh
xx
xxxg
xx
xxf
65
232
2
xx
xxxf
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Homework Questions?
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You-Try #3 (like HW #29-33) Find the horizontal asymptote, if any, of each
function
7
23
7
23
7
234
3
3
3
2
3
x
xxxh
x
xxxg
x
xxxf
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Graphing Rational Functions1. Find the zeros of p(x), the numerator2. Find the zeros of q(x), the denominator3. Identify any vertical asymptotes (numbers that are
zeros of q(x) but not zeros of p(x)). Draw a dashed line.
4. Identify any holes (x-values are numbers that are zeros of both p(x) and q(x))
5. Identify any horizontal asymptotes by examining the leading terms. Draw a dashed line.
6. Find f(-x) to determine if the graph of f(x) has symmetry:
If f(-x)=f(x), then there is y-axis symmetry If f(-x)=-f(x), then there is origin symmetry
xqxp
xf
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Graphing Rational Functions, cont.7. Find the y-intercept by evaluating f(0)8. Identify the x-intercepts (numbers that are
zeros of p(x) but not q(x))9. Pick a few more points to plot10. Draw a curve through the points,
approaching but not touching the asymptotes. If there was a hole identified in step 4, put an open circle at that x-value.
11. Check your graph with a graphing calculator. Remember that it does not properly display asymptotes and holes.
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Example 4 (like HW #37-58) Graph
62
2
xx
xxf
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You-Try #4 (like HW #37-58) Graph
1
3
x
xxf
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Warm-Up: January 18, 2012 Determine any and all asymptotes and holes
of:
3
2
3
62
x
xxxg
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You-Try #5 (like HW #37-58) Graph
1
42
2
x
xxf
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Slant Asymptotes A slant asymptote is a line of the form
y=mx+b that the graph of a function approaches as x±∞
The graph of f(x) has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator
Find the equation of the slant asymptote by division (synthetic or long), and ignore the remainder
xqremainder
bmxxq
xpxf
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Example 7 (like HW #59-66) Find the slant
asymptote and graph
1
12
x
xxxf
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You-Try #7 (like HW #59-66) Find the slant
asymptote and graph
x
xxf
42
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Warm-Up: January 19, 2012 Determine any and all asymptotes and holes
of:
1
452
x
xxxf
16
452
2
x
xxxg
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Applications of Rational Functions The average cost of producing an item
Chemical concentrations over time
Used in numerous science and engineering fields to approximate or model complex equations
n
cnCnC fixed
2
5( )
0.01 3.3
tC t
t
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Example 8 (page 322 #70)The rational function
describes the cost, C(x), in millions of dollars, to inoculate x% of the population against a particular strain of the flu.
a) Find and interpret C(20), C(40), C(60), C(80), and C(90)
b) What is the equation of the vertical asymptote? What does this mean in terms of the variables of the function?
c) Graph the function
x
xxC
100
130
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Assignment Page 321 #1-39 odd, 59, 67